Sparse Grids

Gerstner, Thomas; Griebel, Michael

doi:10.1002/9780470061602.eqf12011

Cited by 19 publications

(19 citation statements)

References 22 publications

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“…Possible schemes for choosing sample points are, e.g., low discrepancy points like Halton points 31 or points that are suitable for polynomial approximations like Chebyshev points. Sparse grid methods 32 combine results computed on a particular sequence of structured grids in order to get the final approximation. This type of technique has also been used together with RBFs 33 .…”

Section: Generation Of Grid Pointsmentioning

confidence: 99%

An adaptive interpolation scheme for molecular potential energy surfaces

Kowalewski

Larsson

Heryudono

2016

The Journal of Chemical Physics

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The calculation of potential energy surfaces for quantum dynamics can be a time consuming task -especially when a high level of theory for the electronic structure calculation is required. We propose an adaptive interpolation algorithm based on polyharmonic splines (PHS) combined with a partition of unity (PU) approach. The adaptive node refinement allows to greatly reduce the number of sample points by employing a local error estimate. The algorithm and its scaling behavior is evaluated for a model function in 2, 3 and 4 dimensions. The developed algorithm allows for a more rapid and reliable interpolation of a potential energy surface within a given accuracy compared to the non-adaptive version.

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Section: Generation Of Grid Pointsmentioning

confidence: 99%

An adaptive interpolation scheme for molecular potential energy surfaces

Kowalewski

Larsson

Heryudono

2016

The Journal of Chemical Physics

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“…to mention all of them. The reader can see the surveys in [6,30,24] and the references therein. For recent further developments and results see in [29,27,28,22,4].…”

Section: Introductionmentioning

confidence: 99%

“…In sampling recovery and numerical integration, a classical model in attempt to overcome it which has been widely studied, is to impose certain mixed smoothness or more general anisotropic smoothness conditions on the function to be approximated, and to employ sparse grids for construction of approximation algorithms for sampling recovery or integration. We refer the reader to [6,24,34,35] for surveys and the references therein on various aspects of this direction.…”

Section: Introductionmentioning

confidence: 99%

“…It can be an efficient tool in some high-dimensional approximation problems, especially in applications ones. Thus, one of the important bases for sparse grid high-dimensional approximations having various applications, are the Faber functions (hat functions) which are piecewise linear B-splines of second order [6,24,29,27,28,22,4]. The representation by Faber basis can be obtained by the Bspline quasi-interpolation (see, e. g., [18]).…”

Section: Introductionmentioning

confidence: 99%

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Sampling and Cubature on Sparse Grids Based on a B-spline Quasi-Interpolation

DźNg

2015

Found Comput Math

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Let X n = {x j } n j=1 be a set of n points in the d-cube, and Φ n = {ϕ j } n j=1 a family of n functions on I d . We consider the approximate recovery of functions f onerror of sampling recovery is measured in the norm of the space L q (I d )-norm or the energy quasi-norm of the isotropic Sobolev space W γ q (I d ) for 1 < q < ∞ and γ > 0. Functions f to be recovered are from the unit ball in Besov type spaces of an anisotropic smoothness, in particular, spaces B α,β p,θ of a "hybrid" of mixed smoothness α > 0 and isotropic smoothness β ∈ R, and spaces B a p,θ of a nonuniform mixed smoothness a ∈ R d + . We constructed asymptotically optimal linear sampling algorithms L n (X * n , Φ * n , ·) on special sparse grids X * n and a family Φ * n of linear combinations of integer or half integer translated dilations of tensor products of B-splines. We computed the asymptotic order of the error of the optimal recovery. This construction is based on B-spline quasi-interpolation representations of functions in B α,β p,θ and B a p,θ . As consequences we obtained the asymptotic order of optimal cubature formulas for numerical integration of functions from the unit ball of these Besov type spaces.

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“…It can be an efficient tool in some high-dimensional approximation problems, especially in applications ones. Thus, one of the important bases for sparse grid high-dimensional approximations having various applications, are the Faber functions (hat functions) which are piecewise linear B-splines of second order [4,25,27,26,28,24,3]. The representation by Faber basis can be obtained by the B-spline quasi-interpolation (see, e. g., [17]).…”

Section: Introductionmentioning

confidence: 99%